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On consistency of a class of estimators for exponential families of Markov random fields on the lattice. (English) Zbl 0787.62100

A statistical estimation problem is considered for parametric exponential non-Gaussian families of Markov random fields, and more generally, Gibbs distributions (GD). Consistency, exponential convergence rates and Bahadur optimality are proved for the maximum and pseudo-maximum likelihood estimators under conditions allowing nonergodic, nonstationary and phase transition cases.
The proofs are based on the large deviations estimates for the GD obtained by the author [C. R. Acad. Sci., Paris, Ser. I 303, 511- 513 (1986; Zbl 0606.60035)], H. Föllmer and S. Orey [Ann. Probab. 16, No. 3, 961-977 (1988; Zbl 0648.60028)] and by S. Olla [Probab. Theory Relat. Fields 77, No. 3, 343-357 (1988; Zbl 0621.60031)].

MSC:

62M40 Random fields; image analysis
60F10 Large deviations
62F12 Asymptotic properties of parametric estimators
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